Let $\kappa$ be an infinite regular ordinal. Define $\Delta_\kappa$ to be the category of ordinals strictly smaller than $\kappa$ (i.e. the ordinals which are elements of $\kappa$, in von Neumann definition), with order-preserving maps as morphisms.

We know that $f$ is injective and order preserving, by that it is easy to prove the claim: let be the greater ordinal between and , clearly we have the two inclusion and , so we can extend to a map (by composing with the embedding of in ).

Clearly is still monotone and injective, being the composition of two monotone and injective morphisms, and by construction we know that , that is is an initial segment of .

We have to prove that for each (i.e. for each ) we have that .We proceed by transfinite induction.

For the claim is trivally true: because every ordinal is great or equal to .

Let assume that for every we have that . Then we have that for each

because of the monotonicity of and the inductive hypothesis.

Now we distinguish between two cases:

if then we have that and so

if is a limit ordinal then for each , hence

whatever the case we always end up having , as we wished to prove.

By transfinite induction it follows that for every .Since has image contained in this proves that for each we have that